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1
Evaluating Formulas on Sparse Graphs
Part 4
Anuj Dawar
University of Cambridge Computer Laboratory
PhD Open, Warsaw, 6 November 2010
Anuj Dawar November 2010
2
Results So Far
1. Tk—the class of graphs of tree-width at most k.
Courcelle (1990) shows that every MSO definable property is decidable in
linear time on this class.
2. Dk—the class of graphs of degree bounded by k.
Seese (1996) shows that every FO definable property is decidable in linear
time.
3. LTWt—the class of graphs of local tree-width bounded by a function t. Frick
and Grohe (2001) show that every FO definable property is decidable in
quadratic time.
4. Mk—the class of graphs excluding Kk as a minor.
Flum and Grohe (2001) show that every FO definable property is decidable
in time O(n5).
Anuj Dawar November 2010
3
Map of Classes
excluded minors bounded local treewidth
planar graphs bounded degree
acyclic graphs
bounded genus
bounded treewidth
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4
Incomparability
D3—the class of graphs of degree 3 does not exclude any minor.
The class of apex graphs does not have bounded local tree-width.
A graph G is an apex graph if there is a vertex v such that G− v is planar.
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Common Generalization
Let t : N → N be a non-decreasing function.
LEMt—the class of graphs G = (V,E) such that for every v ∈ V :
Kt(r) 6≺ NGr (v)
We say that C locally excludes minors if there is some function t such that
C ⊆ LEMt.
Theorem (D., Grohe, Kreutzer)
First-order logic is fixed-parameter tractable on every class C that locally excludes
minors.
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6
Application of Locality Method?
The result would be an easy application of the locality method if we had
established:
There is an algorithm deciding A |= ϕ in time f(l, k)nc
where k is the least value such that Kk 6≺ G
While the Flum-Grohe theorem does give a O(n5) algorithm for the class of
graphs that exclude Kk as a minor, it is not clear if the constants are bounded by
a computable function of k.
Anuj Dawar November 2010
7
Potential Sources of Uncomputability
1. The algorithm decomposing a graph in Mk over the class Lλ,µ relies on a
membership test in a minor-closed superclass of Mk . It is not known
whether the excluded minors for this class are given by a computable function
of k.
2. The algorithm for reducing graphs in Lλ,µ to Lλ relies on membership tests
for Lλ,µ′ (for µ′ ≤ µ) and it is not known if the excluded minors for these
classes are given by a computable function of k.
(D., Grohe, Kreutzer) gives constructive solutions to both these problems.
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Constructive Decomposition
There is a uniform in k algorithm which computes a decomposition of a graph
G ∈ Mk over Lλ,µ in time O(n4).
Instead of clique-sums, the decomposition uses neighbourhood-sums.
(G1, x)⊙x (G2, x)
is the graph obtained by taking the disjoint sum of G1 and G2 while identifying
N1(x) and deleting x.
It is also shown that given G ∈ Lλ,µ, we can effectively find v1, . . . , vµ such
that G \ {v1, . . . , vµ} ∈ Lλ′ for some λ′ computable from λ.
Anuj Dawar November 2010
9
Review
We have gone from graphs of bounded
size to locally excluded minors in four
steps, alternating decomposition steps
with localisation steps.
This seems to be the limit of the
methods. Are there interesting classes
which admit decompositions over
classes with locally excluded minors?
We next look at some other classes
which allow for new methods.
Bounded
Bounded
Bounded
Minors
Minors
Tree-width
Local Tree-width
Size
Excluded
Locally Excluded
Decompositions
Decompositions
Locality
Locality
Anuj Dawar November 2010
10
Map of Classes
excluded minors bounded local treewidth
planar graphs bounded degree
bounded expansion locally excluded minors
acyclic graphs
bounded genus
bounded treewidth
nowhere dense
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11
New Classes
Graph classes of Bounded Expansion were introduced in a series of papers by
Nesetril and Ossona de Mendez.
They are a common generalization of bounded degree and excluded minor
classes.
They were used to study problems of colourability of sparse graphs.
What is a well-behaved notion of sparseness?
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Sparse Classes
For a graph G, we write |G| for the number of vertices in G and ||G|| for the
numer of edges in G.
If G has degree at most d, then ||G|| ≤ d2 |G|.
If G has tree-width at most k, then ||G|| ≤ k|G|.
Theorem (Kostochka)
There exists a constant c such that, for every r, every graph G of average degree
d ≥ c√r log r contains Kr as a minor.
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Sparse Classes
Say a class C of graphs is sparse if there is a c so that for all G ∈ C,
||G|| ≤ c|G|.
Equivalently, C has bounded average degree.
There are pathological sparse classes.
Take the class that contains, for every finite graph G the graph consisting
of the union of G with ||G|| isolated vertices.
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Hereditarily Sparse Classes
Say a class C of graphs is hereditarily sparse if there is a c so that for all G ∈ C,
and every subgraph H ⊂ G we have ||H|| ≤ c|H|.
With any graph G = (V,E), we associate its incidence graph
I(G) = (V ∪ E,F ) where
F = {(v, e) | v ∈ V, e ∈ E and e is incident with v}.
The collection of all incidence graphs is hereditarily sparse but has all the
complexity of the class of all graphs since the map G 7→ I(G) is an easy
first-order reduction.
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Shallow Minors
Recall that H = (U, F ) is a minor of G = (V,E), if we can find a collection of
disjoint, connected subgraphs of G: (Bu | u ∈ U) such that whenever
(u1, u2) ∈ F , there is an edge between some vertex in Bu1and some vertex in
Bu2.
The graphs Bu are called branch sets witnessing that H � G.
If the branch sets can be chosen so that for each u there is b ∈ Bu and
Bu ⊆ NGr (b), we say that H is a minor at depth r of G and write H �r G
Anuj Dawar November 2010
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Bounded Expansion
A class C of graphs is said to have bounded expansion if, for each r, there is a
∇r such that if H �r G for some G ∈ C then ||H|| ≤ ∇r|H|.
In other words, C has bounded expansion if, for every r, the collection of
depth-r minors of graphs in C is sparse.
Mk has bounded expansion by taking ∇r = c√k log k (where c is from
Kostochka’s theorem.
Dk has bounded expansion by taking ∇r = kr .
Anuj Dawar November 2010
17
FO on Bounded Expansion Classes
A very recent result (FOCS 2010) by Dvorak, Kr al and Thomas states:
If C is a class of bounded expansion and ϕ is a first-order sentence, then
G |= ϕ can be decided, for G ∈ C in linear time.
Note: this improves the O(n5) bound for excluded minor classes.
The technique used is quite different to the locality and decomposition techniques
we have seen. It relies on suitable graph colourings.
Anuj Dawar November 2010
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Tree Depth
The tree-depth of a graph G is defined to be the smallest k such that there is
directed forest F of height k and
G is a subgraph of the undirected graph underlying the transitive closure
of F .
For any graph G, tw(G) ≤ td(G),
where tw(G) is the tree-width of G and td(G) is the tree-depth of G.
Anuj Dawar November 2010
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Low Tree-Depth Colourings
Nesetril and Ossona de Mendez prove a remarkable colouring property of classes
of graphs of bounded expansion.
Theorem: (Ne set ril, Ossona de Mendez)
Let C be a class of graphs of bounded expansion. For any p there is an N such
that any graph G ∈ C can be coloured using N colours in such a way that if
C1, . . . , Cp is any set of p colours then G[C1 ∪ · · · ∪ Cp] has tree− depth
at most p.
Moreover, this colouring can be found efficiently—in linear time.
Anuj Dawar November 2010
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Evaluating Existential Formulas
Suppose ϕ is an existential first-order formula.
That is, it is of the form
∃x1 · · · ∃xqθ
where θ is quantifier-free.
If C has bounded expansion, we can evaluate such formulas on graphs in C by
the following process.
Find a colouring of G which guarantees that any q colours induce a graph
of tree-depth at most q.
For each set of q colours, check whether ϕ can be evaluated in the
subgraph induced by these colours.
This establishes that for existential formulas, satisfaction is FPT on C.
Anuj Dawar November 2010
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Quantifier Alternation
To deal with formulas with quantifier alternation Dvorak et al. define a quantifier
elimination procedure.
This requires us to replace, at each step, the graph G with an expansion including
new coloured edges giving the low tree-depth decompositions.
Crucially, they prove that doing this on all graphs in C still gives a graph of
bounded expansion, though with larger bounds.
Anuj Dawar November 2010
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Incomparibility
There are classes C with bounded expansion that do not have locally excluded
minors.
Let C be the collection of graphs G that contain a vertex v such that
G− v is in D3.
There are classes C that locally exclude minors but do not have bounded
expansion.
Lemma (Erdos)
For any d, k ∈ N there is a graph G with girth ≥ k and minimum degree ≥ d.
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Nowhere Dense Graphs
We say that a class C of graphs is nowhere-dense if, for every r, the collection of
graphs
{H | H �r G for some G ∈ C}is not the class of all graphs.
In other words, for each r, there is a Kk that cannot be obtained as a
depth-r minor of any graph in C.
This clearly generalizes bounded expansion classes.
It also generalizes locally excluded minor classes because if Kk �r G then
there is a v in G such that Kk � NG3r+1(v).
Anuj Dawar November 2010
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Trichotomy Theorem
Associate with any infinite class C of graphs the following parameter:
dC = limr→∞
lim supG∈Cr
log ||G||log |G| ,
where Cr is the collection of graphs obtained as minors of a graph in C by
contracting neighbourhoods of radius at most r.
The trichotomy theorem of Nesetril and Ossona de Mendez states that dC can
only take values 0, 1 and 2.
The nowhere-dense classes are exactly the ones where dC 6= 2.
This shows that these classes are a natural limit to one notion of sparseness.
Anuj Dawar November 2010
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FO on Nowhere Dense Classes
It is still an open question whether FO satisfaction is fixed-parameter tractable on
nowhere-dense classes.
Some problems, defined by families of FO formulas, have been shown to be FPT
on such classes.
• Independent Set;
• Dominating Set;
• distance-d dominating set
The proof for these is based on a technique distinct from those we have seen so
far.
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Wide Classes
A set of vertices A in a graph G is said to be r-scattered if for any u, v ∈ A,
Nr(u) ∩Nr(v) = ∅.
Definition
A class of graphs C is said to be wide if for every r and m there is an N such that
any graph in C with more than N vertices contains a r-scattered set of size m.
Example: Classes of graphs of bounded degree.
Non-Example: Trees
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Almost Wide Classes
Definition
A class of graphs C is almost wide if there is an s such that for every r and m
there is an N such that any graph in C with more than N vertices contains s
elements whose removal leaves a r-scattered set of size m.
Example: Trees.
Examples: planar graphs?
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Quasi-Wide Classes
Let s : N → N be a function. A class C of graphs is quasi-wide with margin s if
for all r ≥ 0 and m ≥ 0 there exists an N ≥ 0 such that if G ∈ C and
|G| > N then there is a set S of vertices with |S| < s(r) such that G− S
contains an r-scattered set of size at least m.
We show that any class of nowhere-dense graphs is quasi-wide.
The proof also shows that any class that excludes Kk as a minor is almost wide
with margin k − 2.
Anuj Dawar November 2010
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Finding Scattered Sets
Theorem
There is a computable N so that for any k, r,m ≥ 0, if G is a graph with more
than N(k, r,m) vertices then
1. either Kk �r+1 G; or
2. there is a set S of vertices with |S| ≤ k − 2 such that G− S contains an
r-scattered set of size m.
Anuj Dawar November 2010
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Ramsey’s Theorem
Write [I ]k for the collection of k-element subsets of I .
Theorem (Ramsey)
For each k, l and m, there is an N such that, if |I | > N , and
I1, . . . , Il
is a partition of [I ]k, there is a J ⊆ I with |J | > m, such that [J ]k ⊆ Ii for
some i.
The special case when k = l = 2 is often stated in the following form
For any m there is an N such that any graph with more than N vertices
either contains an m-clique or an m-independent set.
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Expanding Neighbourhoods
Start with the large set W , and take neighbourhoods of increasing radius around
the vertices in W , while thinning out the set so that neighbourhoods are dijoint.
r· · ·
If there are more k neighbourhoods which pairwise have an edge between them,
then Kk �r G.
So, there must be a large number of neighbourhoods that are independent (a
simple application of Ramsey’s theorem).
Anuj Dawar November 2010
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Expanding Neighbourhoods
r-neighbourhoods of vertices in W may also have common neighbours.
r· · ·
We can guarantee that we can find a large subset W ′ ⊆ W which has a
common neighbour.
The number of such common neighbours cannot be k − 1.
Otherwise, we could find Kk−1,k−1 and therefore Kk as a minor (a more
complicated application of Ramsey’s theorem).
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Equivalence
A class C of graphs is uniformly quasi-wide with margin s if for all r ≥ 0 and all
m ≥ 0 there exists an N ≥ 0 such that if G = (V,E) ∈ C and W ⊆ V with
|W | > N then there is a set S ⊆ V with |S| < s(r) such that W contains an
r-scattered set of size at least m in G− S.
Theorem: (Ne set ril, Ossona de Mendez)
The following are equivalent for any class C that is closed under taking induced
subgraphs:
1. C is nowhere dense
2. C is quasi-wide
3. C is uniformly quasi-wide
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Tractable Algorithm
The proof not only gives effective bounds on N , it provides an efficient (in the size
of G) algorithm for finding the bottleneck and the scattered set.
Lemma : Let C be a nowhere-dense class of graphs and h be the function such
that Kh(r) 6�r G for all G ∈ C. The following problem is solvable in time
O(|G|2).
Input: G = (V,E) ∈ C, r,m ∈ N, W ⊆ V with |W | >
N(h(r), r,m)
Problem: compute a set S ⊆ V , |S| ≤ h(r) − 2 and a set A ⊆ W
with |A| ≥ m, such that in G− S, A is r-scattered.
Anuj Dawar November 2010
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Network Centres Problem
The distance-d dominating set problem, also known as the network centres
problem is the following:
Network Centres Problem: Given a graph G and positive integers k, d,
decide whether G contains a set X of k vertices such that every vertex
of G is within distance d of a vertex in X .
We show that this problem is FPT (with parameter k + d) on any class C of
graphs that is effectively nowhere-dense.
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Kernelization
We prove that the following problem is fixed-parameter tractable for any effectively
nowhere-dense class C.
Input: A graph G = (V,E) ∈ C, W ⊆ V , k, d ≥ 0
Parameter: k + d
Problem: Determine whether there is a set X ⊆ V of k vertices which
d-dominates W .
We replace W by a subset W ′ whose size depends only on the parameters, and
such that any X that d-dominates W ′ d-dominates W
This suffices because the above problem is FPT (on all graphs) when
parameterized by k + d+ |W |.
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Algorithm
As long as W is big, we can find a bottleneck S and a large enough scattered set
A ⊆ W in G− S so that there are a1, . . . , ak+2 with the same distance vector
to S.
The distance vector of a is the s-tuple of elements from {0, . . . , d,∞}indicating the distances of a from the elements of s.
If X d-dominates W \ {a1} then it d-dominates W .
Question: Can such a method be generalized to all of first-order logic?
Anuj Dawar November 2010
38
Review
excluded minors bounded local treewidth
planar graphs bounded degree
bounded expansion locally excluded minors
acyclic graphs
bounded genus
bounded treewidth
nowhere dense
For all classes except the last one the picture, we have established that FO
satisfaction is FPT.
For nowhere dense classes this remains an open question.
Techniques deployed use: locality, decompositions, low tree-depth colourings and
wideness.
Anuj Dawar November 2010
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